reserve i, x, I for set,
  A, M for ManySortedSet of I,
  f for Function,
  F for ManySortedFunction of I;
reserve P, R for MSSetOp of M,
  E, T for Element of bool M;

theorem
  P is monotonic implies P..(E (/\) T) c= P..E (/\) P..T
proof
  assume
A1: P is monotonic;
  E in bool M by MSSUBFAM:12;
  then E c= M by MBOOLEAN:1;
  then E (/\) T c= M by MBOOLEAN:14;
  then E (/\) T in bool M by MBOOLEAN:1;
  then
A2: E (/\) T is Element of bool M by MSSUBFAM:11;
  E (/\) T c= T by PBOOLE:15;
  then
A3: P..(E (/\) T) c= P..T by A1,A2;
  E (/\) T c= E by PBOOLE:15;
  then P..(E (/\) T) c= P..E by A1,A2;
  hence thesis by A3,PBOOLE:17;
end;
